An Integrated DFT-FDTD Design of Plasmon-Enhanced Lead-Free $CsSn$$_x$$Ge$$_{1-x}$$I$$_3$ Perovskite LEDs
CsSn$_x$Ge$_{1-x}$I$_3$ as lead-free perovskites are promising for next generation NIR emitting perovskite LEDs due to their tunable bandgaps and stability. However, they suffer from poor light extraction efficiency, and accurate composition-specific optical data for these materials remain scarce. This study presents a DFT-FDTD framework to optimize light extraction via compositional tuning and plasmonic enhancement. First, DFT calculations were performed to obtain composition-specific complex refractive index and extinction coefficient values for $x = 0, 0.25, 0.5, 0.75$, and $1$. Results show bandgap increased from 1.331 eV for CsSnI$_3$ to 1.927 eV for CsGeI$_3$ with increasing Ge content, while refractive index ranges from 2.2 to 2.6 across compositions. These optical constants were then used as inputs for FDTD simulations of a PeLED structure with optimized Au/SiO$_2$ core-shell nanorods for plasmonic enhancement. A 12.1-fold Purcell enhancement was achieved for CsSn$_{0.25}$Ge$_{0.75}$I$_3$, while light extraction efficiency reached 25% for CsSn$_{0.5}$Ge$_{0.5}$I$_3$. LEE enhancement of 36% was obtained for CsSnI$_3$, and spectral overlap between emitter and plasmon resonance reached 96% for Sn-rich compositions. Design guidelines indicate CsSn$_{0.5}$Ge$_{0.5}$I$_3$ offers optimal balance of extraction efficiency (25%), Purcell enhancement (5.3$\times$), spectral overlap (93%), and oxidation stability for wearable and flexible optoelectronic applications, while CsSn$_{0.25}$Ge$_{0.75}$I$_3$ is recommended for applications prioritizing spontaneous emission rate.